Commutator of two subgroups is normal in join

Statement

Suppose $H,K$ are subgroups of a group $G$. Then, the Commutator of two subgroups (?) $[H,K]$ is a normal subgroup of the Join of subgroups (?) $\langle H,K\rangle$.

Facts used

1. Subgroup normalizes its commutator with any subset: If $H\leq G$ is a subgroup and $A$ is a subset of $G$, then $H$ normalizes the following subgroup:

$[A,H]=\langle [a,h]\mid a\in A,h\in H\rangle$

Here, $[a,h]=a^{-1}h^{-1}ah$ is the commutator of the two elements.

Proof

Given: Two subgroups $H,K\leq G$.

To prove: $[H,K]\triangleleft \langle H,K\rangle$.

Proof: By fact (1), $H$ normalizes $[K,H]$, which is the same as $[H,K]$. Also, $K$ normalizes $[H,K]$. Thus, the normalizer of $[H,K]$ in $G$ contains both $H$ and $K$, hence it contains $\langle H,K\rangle$, proving that $[H,K]$ is normal in $\langle H,K\rangle$.